3.3064 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}} \]

[Out]

(2*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/Sqrt[a]

________________________________________________________________________________________

Rubi [A]  time = 0.0975098, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1970, 1357, 724, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x),x]

[Out]

(2*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/Sqrt[a]

Rule 1970

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d^(m + 1), Subst
[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{x}+\frac{c x}{d}}} \, dx,x,\frac{d}{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=4 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \sqrt{\frac{d}{x}}}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.297137, size = 54, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x),x]

[Out]

(2*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/Sqrt[a]

________________________________________________________________________________________

Maple [B]  time = 0.14, size = 94, normalized size = 1.7 \begin{align*} 2\,{\frac{\sqrt{x}}{\sqrt{a}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+c/x+b*(d/x)^(1/2))^(1/2),x)

[Out]

2*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(
d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))/a^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x), x)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

Integral(1/(x*sqrt(a + b*sqrt(d/x) + c/x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x), x)